<s>
Manifold	B-Algorithm
alignment	I-Algorithm
is	O
a	O
class	O
of	O
machine	O
learning	O
algorithms	O
that	O
produce	O
projections	O
between	O
sets	O
of	O
data	O
,	O
given	O
that	O
the	O
original	O
data	O
sets	O
lie	O
on	O
a	O
common	O
manifold	B-Architecture
.	O
</s>
<s>
The	O
concept	O
was	O
first	O
introduced	O
as	O
such	O
by	O
Ham	O
,	O
Lee	O
,	O
and	O
Saul	O
in	O
2003	O
,	O
adding	O
a	O
manifold	B-Architecture
constraint	O
to	O
the	O
general	O
problem	O
of	O
correlating	O
sets	O
of	O
high-dimensional	O
vectors	O
.	O
</s>
<s>
Manifold	B-Algorithm
alignment	I-Algorithm
assumes	O
that	O
disparate	O
data	O
sets	O
produced	O
by	O
similar	O
generating	O
processes	O
will	O
share	O
a	O
similar	O
underlying	O
manifold	B-Architecture
representation	O
.	O
</s>
<s>
By	O
learning	O
projections	O
from	O
each	O
original	O
space	O
to	O
the	O
shared	O
manifold	B-Architecture
,	O
correspondences	O
are	O
recovered	O
and	O
knowledge	O
from	O
one	O
domain	O
can	O
be	O
transferred	O
to	O
another	O
.	O
</s>
<s>
Most	O
manifold	B-Algorithm
alignment	I-Algorithm
techniques	O
consider	O
only	O
two	O
data	O
sets	O
,	O
but	O
the	O
concept	O
extends	O
to	O
arbitrarily	O
many	O
initial	O
data	O
sets	O
.	O
</s>
<s>
Manifold	B-Algorithm
alignment	I-Algorithm
algorithms	O
attempt	O
to	O
project	O
both	O
and	O
into	O
a	O
new	O
d-dimensional	O
space	O
such	O
that	O
the	O
projections	O
both	O
minimize	O
distance	O
between	O
corresponding	O
points	O
and	O
preserve	O
the	O
local	O
manifold	B-Architecture
structure	O
of	O
the	O
original	O
data	O
.	O
</s>
<s>
This	O
is	O
usually	O
encoded	O
as	O
the	O
heat	O
kernel	O
of	O
the	O
adjacency	B-Algorithm
matrix	I-Algorithm
of	O
a	O
k-nearest	O
neighbor	O
graph	O
.	O
</s>
<s>
Finally	O
,	O
introduce	O
a	O
coefficient	O
,	O
which	O
can	O
be	O
tuned	O
to	O
adjust	O
the	O
weight	O
of	O
the	O
'	O
preserve	O
manifold	B-Architecture
structure	O
 '	O
goal	O
,	O
versus	O
the	O
'	O
minimize	O
corresponding	O
point	O
distances	O
 '	O
goal	O
.	O
</s>
<s>
With	O
these	O
definitions	O
in	O
place	O
,	O
the	O
loss	O
function	O
for	O
manifold	B-Algorithm
alignment	I-Algorithm
can	O
be	O
written	O
:	O
</s>
<s>
Solving	O
this	O
optimization	O
problem	O
is	O
equivalent	O
to	O
solving	O
a	O
generalized	O
eigenvalue	O
problem	O
using	O
the	O
graph	B-Algorithm
laplacian	I-Algorithm
of	O
the	O
joint	O
matrix	O
,	O
G	O
:	O
</s>
<s>
The	O
algorithm	O
described	O
above	O
requires	O
full	O
pairwise	O
correspondence	O
information	O
between	O
input	O
data	O
sets	O
;	O
a	O
supervised	B-General_Concept
learning	I-General_Concept
paradigm	O
.	O
</s>
<s>
Project	O
each	O
input	O
data	O
set	O
to	O
a	O
lower-dimensional	O
space	O
independently	O
,	O
using	O
any	O
of	O
a	O
variety	O
of	O
dimension	B-Algorithm
reduction	I-Algorithm
algorithms	O
.	O
</s>
<s>
Perform	O
linear	O
manifold	B-Algorithm
alignment	I-Algorithm
on	O
the	O
embedded	O
data	O
,	O
holding	O
the	O
first	O
data	O
set	O
fixed	O
,	O
mapping	O
each	O
additional	O
data	O
set	O
onto	O
the	O
first	O
's	O
manifold	B-Architecture
.	O
</s>
<s>
Manifold	B-Algorithm
alignment	I-Algorithm
can	O
be	O
used	O
to	O
find	O
linear	O
(	O
feature-level	O
)	O
projections	O
,	O
or	O
nonlinear	O
(	O
instance-level	O
)	O
embeddings	O
.	O
</s>
<s>
Feature-level	O
projections	O
allow	O
any	O
new	O
instances	O
to	O
be	O
easily	O
embedded	O
in	O
the	O
manifold	B-Architecture
space	O
,	O
and	O
projections	O
may	O
be	O
combined	O
to	O
form	O
direct	O
mappings	O
between	O
the	O
original	O
data	O
representations	O
.	O
</s>
<s>
Manifold	B-Algorithm
alignment	I-Algorithm
is	O
suited	O
to	O
problems	O
with	O
several	O
corpora	O
that	O
lie	O
on	O
a	O
shared	O
manifold	B-Architecture
,	O
even	O
when	O
each	O
corpus	O
is	O
of	O
a	O
different	O
dimensionality	O
.	O
</s>
<s>
Manifold	B-Algorithm
alignment	I-Algorithm
also	O
facilitates	O
transfer	B-General_Concept
learning	I-General_Concept
,	O
in	O
which	O
knowledge	O
of	O
one	O
domain	O
is	O
used	O
to	O
jump-start	O
learning	O
in	O
correlated	O
domains	O
.	O
</s>
<s>
Applications	O
of	O
manifold	B-Algorithm
alignment	I-Algorithm
include	O
:	O
</s>
<s>
By	O
representing	O
documents	O
as	O
vector	O
of	O
word	O
counts	O
,	O
manifold	B-Algorithm
alignment	I-Algorithm
can	O
recover	O
the	O
mapping	O
between	O
documents	O
of	O
different	O
languages	O
.	O
</s>
